Spring 2014

April 3, 2014

Veech Surfaces — Ian Frankel

We will discuss the relationship between SL(2,R)-orbits of
translation surfaces and the dynamics of the straight line flows on such
surfaces and the horocycle flow on the moduli space of quadratic
differentials. Of special interest will be Veech surfaces that is, those
with closed orbits.

April 10, 2014

A new of proof of Bowen Rigidity for limit sets of convex-cocompact surface groups — Andy Sanders

In 1979, Rufus Bowen proved that the Hausdorff dimension of the limit set of a quasi-Fuchsian group is equal to one if and only if the group is Fuchsian. This has been greatly generalized by Bonk and Kleiner to quasi-convex, cocompact group actions on CAT(-1) metric spaces. Using techniques from Riemannian geometry and smooth dynamical systems, we give a new proof of these results in the case of a convex-cocompact surface group acting on a CAT(-1) Riemannian manifold X. These are achieved by a new lower bound on the Hausdorff dimension of the limit set of such a group in terms of the data of an equivariant, minimal surface in X. In the case that X is hyperbolic 3-space, this estimate gives a quantification of Bowen’s theorem along the following lines: Hausdorff dimension near one implies “close” to Fuchsian.

April 17, 2014

Complex Projective Structures and Grafting — Sebastian Hensel

We all know that closed surfaces of genus at least 2 carry hyperbolic structures. We also like to put singular flat structures on them to do Teichmüller theory. But there is actually a third type of structure — complex projective structures — which have a very interesting (and mysterious!) interplay both with geometry and analysis of Riemann surfaces. Also, Thurston liked them, so who are we to argue?
In this talk, I’ll explain what they are, what they are connected to and hopefully convince you to find them interesting as well!

Massey products are higher-order multiplications in cohomology that generalize the cup product. In this talk we will define Massey products and indicate some of their applications in geometric topology. We will then explain the relationship between Massey products and the lower central series of a group. As an application we will give a brief sketch of a hypothetical approach to proving the Poincaré conjecture by means of a deep analysis of the Johnson filtration of the Torelli group.

How does one tell the difference between two collections of curves? — Jonah Gaster

Given a pair of collections of disjoint simple closed curves on a closed oriented surface, it is straightforward to check when they have the same orbits under the mapping class group using the topology of the cut open surface and the dual graphs to the systems of curves. When the curve systems have intersections, the dual graph may not be well-defined. We will explore a useful fix for this problem, Sageev’s dual cube complex, which we show characterizes mapping class group orbits for filling systems of curves. We will use this invariant to give new bounds for the number of mapping class group orbits of maximal complete 1-systems of curves on a surface. This is joint work with Tarik Aougab.

May 8, 2014

The saga of strong approximation for thin groups — Wouter van Limbeek

Lattices in semisimple groups have been widely studied through and for
geometry, topology, number theory, group theory and dynamics. Under the assumption of arithmeticity, an enjoyable phenomenon is strong interplay (called “strong approximation”) between phenomena at finite primes and globally. On the other hand, lattices are not so easy to come by, but no worries — amazingly, strong approximation holds more generally for so-called thin groups. I will try to give a sketch of some of the ideas in establishing “strong approximation”, and why one might care. Time permitting I will discuss the Lubotzky alternative, which can be interpreted as a strong restriction on the structure of arbitrary (!) finitely generated linear groups.

May 15, 2014

A foray into iterated loop space theory — Jenny Wilson

This talk will be an informal survey of some topics in algebraic topology: loop spaces, iterated loop spaces, generalized cohomology theories, spectra, and operads. For some motivation, consider the following question — given a CW-complex , how can we determine whether has the homotopy type of a (based) loop space ?

May 22, 2014

Surprises from geometric group theory — Katie Mann

Warm up problem: Prove that no finite collection of convex heptagonal tiles can be used to tile the plane. (we’re not assuming the tiling is periodic. For example, below is a nice picture of a way to tile the plane with finitely many (two in this case) convex quadrilateral tiles).

Actual talk: Small cancellation theory! Random groups! A proof that “the generic group is a Burnside group” (finitely generated, infinite, but with every element finite order) (yikes!) (!) Annnnnd, finally, how these things are all secretly related. Including the warm-up problem.

[reference: other than the warm-up, most of the content of this talk comes from Ghys’ excellent paper “Groupes Aléatoires d’après Misha Gromov”]

May 29, 2014

Symmetry gaps and minimal orbifolds — Wouter van Limbeek

In 1893 Hurwitz proved that a hyperbolic surface of genus has isometry group of order at most . Are similar estimates possible for other metrics on such a surface? Other hyperbolic manifolds? Locally symmetric spaces? Aspherical manifolds? I will discuss some answers to these questions, and describe the following generalSymmetry gap theorem. A closed manifold either admits an action by a compact connected Lie group or the order of the isometry group can be bounded only in terms of some geometric quantities (dimension, curvature bounds, etc.).

May 29, 2014

\pi_1(M) does not act on M by diffeomorphisms* — Bena Tshishiku

This talk** will be about group actions and characteristic
classes. Pick your favorite infinite group G, and your favorite closed
manifold X. Ask yourself, “Does G act on X?” If you answer “Yes!”, then choose a different example. Now ask yourself, “How might I show that G does not act on X?” I will discuss a special case of this problem (see the title) and an approach using characteristic classes.

* Come to the talk for a precise statement.
** This is a practice talk. If you choose to come, you will be rewarded with
something sweet to eat.